64 CHAPTER 1 Sets, Proof Templates, and Induction
(c) Fo + F 3 + + F 3 = F 3 n+ 2 /2 for n > 0
(d)F 1 =Fn " - (-1)n forn >_ 0
- The Lucas numbers are defined as LO = 2, L, = 1, and Ln = Ln-1 + Ln-2 for n >
2. Prove the following identities for Lucas numbers.
(a) Lj + L 2 + .. • + Ln = Ln+ 2 - 3 for n > 1
(b) L2^2 2 ... 2-forn>2
(c) L 2 + L 4 + - "- +L 2 n = L 2 n+l - 1 for n > 2- Find the value of the following sums:
(a) 2 + 2 - 3n
(b) I-1+1-+ +(In
(c) -2 + 4- 8 +^16 + ... + (-2)11
(d) 1.03 + (1.03)2 + (1.03)3 +-... + (1.03)n - Find a rational number representing each of the following repeating decimals:
(a) 0.537537537537537537537537537...
(b) 31.25469696969696969696969... - A fixed dose of a given drug increases the concentration of that drug above nor-
mal levels in the bloodstream by an amount Co (measured in percent). The effect
of the drug wears off over time such that the concentration at some time t is Coe-kt
where k is the known rate at which the concentration of the drug in the bloodstream
declines.
(a) Find the residual concentration R, the accumulated amount of the drug above nor-
mal levels in the bloodstream, at time t after n doses given at intervals of to hours
starting with the first dose at t = 0.
(b) If the drug is alcohol and 1 oz. of alcohol has Co = 0.05%, how often can a "dose"
be taken so that the residual concentration is never more than 0.15%? Assume
k = (1/3) ln(2).- (a) Prove by induction that 2n > n for all n > 0.
(b) Prove that 2n > n directly from Theorem 2 in Section 1.7.4, without explicit use of
induction. (That is, Theorem 2 in Section 1.7.4 itself was proved using induction,
but you should not have to do any additional induction.)
(c) Prove by induction that 2" > n^3 for n > 10. - Prove by induction:
(a) There is a natural number k such that n! > n^3 for all n > k. (Try to find the least
such number k.)
(b) n! > n^4 for n > 7.
31. Let T = {n E N : sin(n •7r) = 0}. Prove that T = N. (Hint: sin(a + b) = sin(a) •
cos(b) + cos(a) , sin(b).)- Prove assertion 1 from Lemma 1.
- (a) Suppose you take out a mortgage for A dollars at a monthly interest rate I and
a monthly payment P. (To calculate I: if the annual interest rate is 12%, divide
by 12 to get a monthly rate of 1%, then replace the percentage with the decimal
fraction 0.01.) Let An denote the amount you have left to pay off after n months.
So, A 0 = A by definition. At the end of each month, you are first charged interest